The area of a right triangle is 18√3. One of the acute angles is 30 °. Find the length of the leg opposite this corner.

1. The ratio of the legs BC and AC of a right-angled triangle ABC with an acute angle ∠A = 30 ° is equal to the tangent of this angle:

BC: AC = tg30 ° = √3 / 3.
BC = √3 / 3 * AC;
AC = √3 * BC.
2. The area of a right-angled triangle is equal to half the product of the legs:

S = 1/2 * AC * BC;
S = 1/2 * √3 * BC * BC;
S = √3 / 2 * BC ^ 2.
3. Substitute the area value in this equation and find the leg BC:

BC ^ 2 = 2S / √3;
BC ^ 2 = 2 * 18√3 / √3;
BC ^ 2 = 36;
BC = 6.
Answer: 6.